Management of a Shared Spectrum Network in Wireless Communications

Transcription

1 Maagemet of a Shared Spectrum Network i Wireless Commuicatios Shiig Wu, Jiheg Zhag, Rachel Q. Zhag Departmet of Logistics ad Maritime Studies The Hog Kog Polytechic Uiversity, Hog Kog Departmet of Idustrial Egieerig ad Logistics Maagemet The Hog Kog Uiversity of Sciece ad Techology, Hog Kog s.wu@polyu.edu.hk, jiheg@ust.hk, rzhag@ust.hk We cosider a bad of the electromagetic spectrum with a fiite umber of idetical chaels shared by both licesed ad ulicesed users. Such a etwork differs from most may-server, two-class queues i service systems icludig call ceters due to the restrictios imposed o the ulicesed users i order to limit iterferece to the licesed users. We first approximate the key performace idicators, amely the throughput rate of the system ad the delay probability of the licesed users uder the asymptotic regime, which requires the aalysis of both scaled ad uscaled processes simultaeously usig the averagig priciple. Our aalysis reveals a umber of distictive properties of the system. For example, sharig does ot affect the level of service provided to the licesed users i a asymptotic sese eve whe the system is critically loaded. We the study the optimal sharig decisios of the system to maximize the system throughput rate while maitaiig the delay probability of the licesed users below a certai level whe the system is overloaded. Fially, we exted our study to systems with time-varyig arrival rates ad propose a diffusio approximatio to complemet our fluid oe. Key words : spectrum maagemet; may-server queues; fluid approximatio; averagig priciple History : Mauscript OPRE Itroductio The radio spectrum refers to the rage of frequecies suitable for wireless commuicatios i televisio ad radio broadcastig, aviatio, public safety, cell phoes, ad so o. Util recetly, spectrum regulatory bodies icludig the Federal Commuicatios Commissio (FCC) i the US ad the Europea Telecommuicatios Stadards Istitute (ETSI) have always allocated spectrum bads exclusively to certai service providers whose users are referred to as primary or licesed users, ofte based o the radio techologies available at the time of allocatio. Such static spectrum allocatio mitigates iterferece to essetial services, yet it creates uderutilizatio of the allocated spectrum, which ca be below 20% eve durig high demad periods i certai geographic areas. For istace, durig the high demad period of a political covetio held i New York City i 2004, oly about 3% of the allocated spectrum was utilized (Prasad et al. 200). Studies coducted by the FCC, uiversities, ad idustry also revealed that a major part of the spectrum is ot fully

2 2 utilized most of the time. O the other had, over the past decades, the covergece of voice ad data i wireless commuicatios triggered by the covergece of wireless ad Iteret techologies has led to a explosio i the umber of bits trasmitted over the air (Biglieri et al. 203). Sice it is usually difficult to ope up higher frequecy bads for mobile applicatios as trasmissio becomes less reliable i those bads, the existig radio spectrum for data trasmissio is reachig its capacity. A atural approach to alleviate the artificial scarcity of spectrum due to static allocatio is to allow opportuistic use of temporarily idle chaels by ulicesed or secodary users to icrease the throughput of already allocated spectrum. This is referred to as opportuistic spectrum access (Hossai et al. 2009). However, allowig ulicesed users access may cause iterferece to existig licesed users. Thus, such paradigm of operatio requires () the kowledge of the state of frequecy bads (e.g., chael availability, queues, etc.) i real time ad (2) a effective cotrol mechaism to gover spectrum usage by ulicesed users, which led to the developmet of the cocept of cogitive radio, first itroduced by Mitola ad Maguire (999). Usig advaced radio ad sigal processig techology, cogitive radio is a software-defied radio device that ca itelligetly sese ad explore the spectrum eviromet, track chages, commuicate iformatio amog differet trasceivers ad react accordig to a cotrol mechaism (Hossai et al. 2009). It is widely regarded as oe of the most promisig techologies for future wireless commuicatios ad may potetially mitigate, through dyamic spectrum access, the problem of radio spectrum scarcity. It is obvious that implemetatio of a cogitive radio etwork ivolves both techological ad operatioal issues, yet much of the research is focused o the former (see Sectio 2. for some relevat literature). I this paper, we focus o the operatioal issues by cosiderig a bad of spectrum with multiple idetical chaels shared by both licesed ad ulicesed users. Sice the spectrum has already bee allocated to the licesed users ad it is usually difficult to set aside a subset of chaels for either groups i reality for techical reasos, we assume all the chaels are accessible by both licesed ad ulicesed users as i most existig literature i electrical egieerig. Furthermore, although cocurret trasmissio is allowed i some etworks uder which the mai cocer is techological (e.g., the power level at which a ulicesed user is allowed to trasmit), we focus o systems where each chael serves oly oe user at a time, referred to as the iterweave paradigm (Biglieri et al. 203). Thus, the etwork cosidered is a two-class queue served by a sigle pool of homogeeous servers as i applicatios i service systems such as call ceters ad healthcare but with some distictive features due to the restrictios imposed o the ulicesed users (Hossai et al. 2009). () Whe all the chaels are occupied upo arrival, a licesed user will joi a queue alog with other waitig licesed users who will be served first-ifirst-out (FIFO) as soo as a chael becomes available, while a ulicesed user will joi a queue

3 3 alog with other waitig ulicesed users ad will oly be allowed to sese chael availability periodically. A ulicesed user ca oly occupy a chael whe a available chael is detected ad o licesed users are waitig, ad may also abado the system every time he seses but fids o available chael. Such a queue where users wait for retrial is referred to as a orbit queue i the queueig literature ad is commo i computer ad commuicatios etworks (Artalejo ad Gómez-Corral 2008). (2) Whe i trasmissio, a licesed user ca trasmit util his service requiremet is fulfilled, while a ulicesed user is oly allocated a fixed amout of time, referred to as a service sessio, approachig the ed of which he has to stop trasmissio to sese the eviromet as sesig caot occur simultaeously with data trasmissio. He will be allowed to cotiue for aother service sessio oly if he seses o waitig licesed users. Otherwise, he has to release the chael ad joi the orbit queue alog with other ulicesed users or abado the system if he eeds more time. Note that data trasmissio ca be iterrupted ad resumed, hece more complicated cotrol policies tha those i call ceters are allowed, which leads to ew maagerial isights. Assumig that perfect sesig ca be achieved i a fixed amout of time ad both licesed ad ulicesed users arrive accordig to Poisso processes, we first perform i-depth aalysis o the key performace idicators i the maagemet of shared spectrum etworks, amely the delay probability of the licesed users ad the system throughput rate. We the focus o the restrictios that eed to be imposed o the ulicesed users whe i service ad waitig, i.e., the legth of a service sessio ad the sesig frequecy while waitig. Ituitively, the loger a service sessio is, the less sesig a ulicesed user eeds to perform ad hece a higher system throughput rate. Yet, loger service sessios ca cause more iterferece to the licesed users. Likewise, the more frequetly a ulicesed user seses chael availability while waitig, the sooer he is able to fid a available chael but the more iterferece he causes to the licesed users. Thus, there is a tradeoff betwee the throughput rate ad the level of iterferece to the licesed users whe decidig o the legth of a service sessio ad the sesig frequecy. The goals of this research are to aswer the followig questios: () Should a give bad of spectrum be shared with ulicesed users? (2) Whe sharig is permitted, how log should ulicesed users be allowed to trasmit each time they occupy a chael ad how frequetly should they be allowed to sese chael availability while waitig? (3) Uder what coditios is sharig more beeficial? (4) How will the decisio chage with ucertai arrival rates or time-varyig arrivals? Sice the bad of spectrum cosidered usually cosists of hudreds or thousads of chaels, we ca treat the system as a large etwork, ad approximate the performace uder the asymptotic regime as i Gupta ad Kumar (2000) ad El Gamal et al. (2006). Due to the restrictios imposed o the ulicesed users whe i service ad waitig, we eed to aalyze both scaled ad uscaled

4 4 processes simultaeously usig the averagig priciple, i.e., approximatig the uscaled process by its log-ru average. We the formulate the problem as fidig the optimal restrictios o the ulicesed users to maximize the throughput rate while maitaiig the delay probability of licesed users below a certai level. Our mai fidigs are as follows:. Sesig frequecy of the ulicesed users while waitig: Surprisigly, sesig frequecy does ot affect the system performace asymptotically as log as the ulicesed users are required to sese chael availability, which takes time ad prevets them from occupyig idle chaels istataeously. Thus, there is o eed to impose ay restrictio o the sesig frequecy from the operatioal perspective. The decisio thus should primarily be based o techological cocers, for istace, power cosumptio associated with each sesig activity. 2. The legth of a service sessio: Ituitively, shorter service sessios should cause less iterferece to ad hece lower the delay probability of the licesed users. However, with shorter service sessios, the ulicesed users eed more service sessios to fiish their service ad hece eed to perform more sesig activities while occupyig a chael. Thus, shorter service sessios do ot always improve the delay probability. 3. Optimal sharig decisios: Whe the system is uder or critically loaded, the iterferece of the ulicesed users to the licesed users is egligible ad there is o eed to impose a restrictio o the service process of the ulicesed users either. That is, allowig the ulicesed users to complete their trasmissios without restrictio will ot cause ay iterferece to licesed users asymptotically as the delay probability is 0. This result is very differet from that of most o-preemptive queueig systems uder which the delay probability is strictly betwee 0 ad whe the system is critically loaded. Whe the system is overloaded, the delay probability of the licesed users is quasi-covex i the legth of the service sessios of the ulicesed users, strictly betwee 0 ad ad icreasig i the load. Thus, a restrictio o the service process of the ulicesed users should be imposed oly whe the load is above a threshold. Furthermore, a shorter service sessio should be allocated as the load icreases util spectrum sharig is o loger feasible. The isight that it is possible to improve spectrum utilizatio while guarateeig a very high service level, expected by licesed users i practice, is very ecouragig ews. Thus, spectrum sharig ca potetially be a socially optimal solutio to alleviatig spectrum scarcity. 4. For a give system load, a shorter service sessio should be allocated to the ulicesed users () as the proportio of the licesed users icreases, (2) if there are fewer licesed users with loger service times, or (3) if there are more ulicesed users with shorter service times. As the service sessio shortes, more ulicesed users will abado the system, which lowers the throughput rate uder scearios () ad (2). Therefore, spectrum sharig is beeficial to

5 5 systems with a smaller proportio of licesed users or a large umber of licesed users with shorter service times. 5. Whe the arrival rates are time varyig, a shorter service sessio should be allocated to the ulicesed users durig busy periods. Although optimal cotrol requires cotiuous adjustmet i real time, ear-optimal cotrol ca be accomplished with occasioal adjustmets. To the best of our kowledge, this is the first comprehesive study of a shared etwork i wireless commuicatios. Although there have bee some attempts by researchers i electrical egieerig usig relatively simple queueig models, our model captures may more of the features of such a system. We are able to ucover complicated system dyamics ad obtai maagerial isights differet from those draw from the may well-studied service systems. Our work ot oly opes the door for ew applicatios of existig queueig theory i wireless commuicatios, but may also stimulate the developmet of ew methodologies. The remaider of this paper is orgaized as follows. We review the relevat literature i both electrical egieerig ad queueig theory i the ext sectio ad describe the problem of dyamic spectrum sharig i detail i Sectio 3. I Sectio 4, we provide a fluid approximatio ad study the optimal sharig decisios of the system. I Sectio 5, we offer the ituitio behid the costructio of the fluid model ad give justificatios for the fluid approximatio. We exted our aalysis to systems with time-varyig arrival rates ad discuss a diffusio scaled approximatio i Sectio 6. We coclude our paper ad provide some future research directios i Sectio 7. The proofs ca all be foud i the Appedix. 2. Literature Review I this sectio, we will first provide some backgroud o the research o opportuistic spectrum access, mostly i electrical egieerig. Sice we will model a shared etwork as a multi-class, may-server queue where the ulicesed users joi a orbit queue ad aalyze it usig the averagig priciple, we will review the relevat literature i queueig theory ad its applicatios. 2.. O Opportuistic Spectrum Access Most of the work o opportuistic spectrum access focuses o the techological issues such as the sesig techology to detect idle chaels (Mishra et al. 2006), sigal ecodig (Devroye et al. 2006) ad the cotrol of the trasmit power to limit iterferece (Basal et al. 2008). For research o various techological issues associated with cogitive radio, readers may refer to Akyildiz et al. (2006) ad Goldsmith et al. (2009). Research o the operatioal issues uder simplified settigs, however, remais scat. Huag et al. (2008) perform a aalytical study o a sigle-chael system with oe licesed ad oe

6 6 ulicesed user, as well as umerical studies o a multi-chael system. They also cosider the decisios o the sesig frequecy of ulicesed users ad how log ulicesed users should be allowed to trasmit i their umerical study. Zhao et al. (2008) study the optimal access strategy of a ulicesed user based o the sesig outcome give that each chael has already bee assiged to a specific licesed user, while Capar et al. (2002) compare the system performace i terms of badwidth utilizatio ad blockig probability whe a licesed user ca be assiged to ay chael radomly or i a cotrolled way. For a more comprehesive picture of the various issues i dyamic spectrum maagemet ad cogitive radio etworks, readers may refer to Hossai et al. (2009) ad Biglieri et al. (203) O Queueig Theory ad Applicatios Multi-Class, May-Server Queues Sice a bad of spectrum cosists of hudreds or thousads of chaels ad there are both licesed ad ulicesed users, the literature of multi-class, may-server queues is relevat. The study of may-server queues was substatiated by the semial work of Halfi ad Whitt (98), who derive the steady-state distributio of the diffusio limits ad establish the square root law describig the relatioship betwee the system load ad delay probability. The mathematical isights of the square root law have sice bee exteded ad widely adopted i the daily maagemet of call ceters aroud the world. Later, Puhalskii ad Reima (2000) exted the study to multi-class models. There is a large body of work o multi-class, may-server systems due to their applicatios i call ceters, maufacturig ad computer-commuicatio systems with a focus o asymptotic optimal cotrol of the uderlyig systems. For example, Atar et al. (2004) study asymptotic optimal schedule policies, Gurvich ad Whitt (2009) propose a family of queue-ad-idleess-ratio rules for routig ad schedulig, ad Maglaras ad Zeevi (2004, 2005) examie the pricig, capacity sizig ad admissio cotrol decisios i a differetiated service system with guarateed (high priority) ad best-effort (low priority) users. Our model differs from the existig work i that the service (i.e., data trasmissio) of the ulicesed users may be fulfilled after multiple iterruptios, which is ot the case i most other applicatios. Sice the service of ulicesed users may be iterrupted by waitig licesed users, the literature o queues with service iterruptio caused by preemptive priority, which dates back to White ad Christie (958) i sigle server settigs, is also relevat. For a review o some of the early work, we refer the reader to Jaiswal (968). Amog the existig work, most focuses o characterizig the steady state distributios of the queue legth, the sojour time ad so o for a give priority disciplie. For example, Brosh (969) derives the expressios for the expected time from arrival to iceptio of service ad provides bouds for the expected sojour time for each class whe

7 7 all classes have the same service rates. Buze ad Bodi (983) obtai the exact expressios for the mea sojour times whe all classes have the same service rates ad provide approximatios whe differet classes have differet service rates. Recetly, Wag et al. (205) coduct the exact aalysis of the steady state of a preemptive M/M/c queue whe differet classes have differet service rates. I our paper, we focus o the cotrol of the service process of the ulicesed users, i.e., how their service processes should be iterrupted. Orbit Queues Sice the ulicesed users joi a orbit queue i our settig, the literature alog this lie is also relevat. Yag ad Templeto (987) ad Fali ad Templeto (997) offer a survey ad a comprehesive summary of the earlier papers, respectively. Later, Madelbaum et al. (2002) provide a aalytical approximatio to the key performace of a may-server queueig system with abadomet ad retrials uder a asymptotic regime. I all these papers, eve though customers may joi a orbit queue for retrial if they caot be served immediately upo arrival, their service caot be iterrupted oce started. Recetly, a umber of studies cosider systems where customers may require repeat service due to uresolved or ew issues. For istace, de Véricourt ad Zhou (2005) ad Zha ad Ward (204) study a customer-routig problem i call ceters with callbacks, while de Véricourt ad Jeigs (2008) ad Yom-Tov ad Madelbaum (204) examie a staffig problem for membership services ad healthcare systems where customers may require multiple rouds of service. These systems differ from ours i that customers will wait i a FIFO queue for retrial if the systems are busy upo arrival, although they will first joi a orbit queue after they have had a roud of service. Allowig the ulicesed users to retry ad joi a orbit queue as i our settig sigificatly complicates the aalysis sice there may be a large umber of customers switchig frequetly betwee beig i service ad beig i the orbit queue. The Averagig Priciple Oly a few studies i the queueig literature have required the use of the averagig priciple. Buildig o a fudametal theory of the averagig priciple by Kurtz (992), Hut ad Kurtz (994) study martigales ad related radom measures of large loss etworks. Whitt (2002) summarizes the early studies o schedulig multi-class queues usig the averagig priciple. Recetly, a series of studies by Perry ad Whitt (20a,b, 203) has applied the averagig priciple to obtai both the fluid ad diffusio limits for a overloaded X model of may-server queues, ad to derive isights about the asymptotic optimal cotrol of the system. Pag ad Perry (205) apply the averagig priciple to obtai a logarithmic safety staffig rule for call ceters with call bledig. We adopt some of the methodologies developed by Hut ad Kurtz (994) ad Perry ad Whitt (20a).

8 8 3. Problem Descriptio ad Assumptios 3.. The Sharig Network ad Performace Measures We cosider a bad of spectrum cosistig of idetical chaels shared by both licesed ad ulicesed users, deoted as user types ad 2 respectively, ad each chael ca oly be occupied by oe user at a time. That is, cocurret trasmissio is ot allowed. Furthermore, we assume that perfect sesig ca be achieved i a fixed amout of time µ s by a ulicesed user. Type i users arrive accordig to a Poisso process with the rate λ i ad require a expoetial amout of service time with the rate µ i, i =, 2. If there is a available chael, a arrivig licesed user will occupy it immediately util his service requiremet is fulfilled. Otherwise, he will joi a queue alog with other waitig licesed users who will be served FIFO as the chaels become available. Next, we describe the service ad waitig processes of the ulicesed users i the shared etwork illustrated i Figure where I (t) is the umber of idle chaels ad Q i (t) is the queue legth of type i users at time t. Accordig to the policy, Q (t)i (t) = 0. () Upo arrival, a ulicesed user will occupy a chael if there is oe available. Otherwise, he will joi a orbit queue alog with other waitig ulicesed users with probability φ or abado the system. The service process: Oce he occupies a chael, a ulicesed user is allocated a fixed amout of uiterrupted time, referred to as a service sessio (Liu ad Wag 200), regardless of his service requiremet. If he eeds more time ad fids o licesed user waitig at the ed of a sessio through sesig, he is allowed to cotiue for aother service sessio. Sice sesig caot occur simultaeously with data trasmissio ad must be iterweaved, he eeds to devote the last µ s amout of time i each service sessio to sese the eviromet if he eeds more time. Hece, we deote the legth of a service sessio by µ t + µ s where µ t is the amout of time allowed for trasmissio i a service sessio. If the ulicesed user completes his trasmissio withi µ t amout of time i a sessio, he will release the chael without sesig ad leave the system. Otherwise, he will have to sese the eviromet ad his service will be iterrupted if he fids a waitig licesed user, i which case he will joi the orbit queue with probability φ or abado the system. The waitig process: While waitig i the orbit queue, a ulicesed user will oly be allowed to sese chael availability periodically. Let deote the time betwee sesig activities, θ which icludes the time eeded for sesig chael availability. After each sesig activity, he will occupy a chael if he fids a idle oe. Otherwise, he will abado the system with probability φ, or stay i the orbit queue for aother sesig activity with probability φ.

9 9 As oe ca see, the etwork has its distictive characteristics which are ot preset i most existig multi-class, may-server queueig systems due to the restrictios o the service ad waitig processes of the ulicesed users, i.e., the trasmissio time µ t i a service sessio ad the sesig frequecy θ. The data trasmissio of the ulicesed users ca be iterrupted ad resumed for ay umber of times ad sesig for chael availability by the ulicesed users i the queue is oly allowed periodically. As a result, a ulicesed user may abado the system upo arrival, after spedig some time i the queue without receivig ay service, or after receivig partial service. Furthermore, each ulicesed user i the orbit queue eeds to sese chael availability idepedetly, which guaratees certai idleess i the system eve whe there are waitig ulicesed users. These features are ew i the queueig literature ad iterestig, yet sigificatly complicate the aalysis. Chaels λ Queue I (t) > 0 λ 2 I (t) = 0 φ φ Abado Wait & Sese φ I (t) > 0 I (t) = 0 eed more service φ Q (t) = 0 Abado φ Q (t) > licesed users ulicesed users φ Abado Figure The spectrum sharig etwork The performace measures we are cocered with are the throughput rate of the system ad the probability that all the chaels are occupied upo the arrival of a licesed user, referred to as the delay probability. The goal is to fid the trasmissio time µ t i a service sessio ad the sesig frequecy θ of the ulicesed users that maximize the throughput of the ulicesed users while guarateeig the delay probability of the licesed users below a certai level.

10 Modelig Assumptios Sice the problem is aalytically itractable, we will first approximate the determiistic trasmissio time, sesig time, ad the time betwee cosecutive sesig activities i the orbit queue by the expoetial distributios with the same meas. Table presets a simulatio study of the delay probability of the licesed users ad the throughput rate of the ulicesed users with determiistic times ad expoetial times whe λ = 0.2, λ 2 = 0.9, = µ 2 =, µ s = 0.00, θ = 0.4, ad φ = 0.5. For µ t {, 0.6, 0.2}, we set = 00, 500, 000, 2000, 4000 ad let λ i = λ i. We report the meas ad 0.95 cofidece itervals of the delay probabilities ad throughput rates. As oe ca see, approximatig the determiistic times by the expoetial times does ot reduce the accuracy very much, especially whe is large as i our applicatio where is i the hudreds or thousads. µ t Table delay probability throughput rate Determiistic Expoetial Determiistic Expoetial ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.0004 Fluid ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.0003 Fluid ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.0004 Fluid Compariso of performace measures with determiistic vs. expoetial times. With the expoetial times metioed above, the probability that a ulicesed user will complete his trasmissio i a service sessio is give by p = µ 2 µ 2 +µ t. Furthermore, the actual amout of time a ulicesed user will occupy a chael i each service sessio follows a phase-type distributio with mea µ = + ( p) = µ t + µ s, (2) µ 2 + µ t µ s (µ 2 + µ t )µ s which is less tha the allocated sessio time µ t + µ s. If we let Z i (t) deote the umber of chaels occupied by type i users at time t, the istataeous throughput rate at time t is give by pµz 2 (t).

11 With hudreds or thousads of chaels i a bad of spectrum, performig a aalytical study of the shared etwork uder a large system scalig to be defied below is ot oly for techical tractability, but also appropriate. Defiitio (Asymptotic Regime). There exist positive real umbers λ i, i =, 2, such that Here λ i λ i lim = λ i, ad λ <. represets the size of type i users. Differet cogitive radio etworks have differet proportios of licesed ad ulicesed users. I IEEE WRANs, ulicesed users outumber licesed users (Zhag et al. 2009, Jia et al. 2008), i.e., λ 2 > λ, while i TV white space etworks, licesed users are the majority (va de Beek et al. 202), i.e., λ > λ 2. I Gog et al. (205), the licesed users (from a dow-lik cellular system) ad the ulicesed users (from a ad hoc etwork) have comparable umbers, i.e., λ λ 2. Uder the asymptotic regime, we will add a bar to the existig otatio to represet the scaled processes i our model, e.g., Q i (t) = Q i (t), ad use the lower case, e.g., q i(t), to represet the correspodig fluid model, which will be prove to be the fluid limit of the scaled processes. 4. Mai Results ad Isights Uder the asymptotic regime, the processes ivolved are scaled ad the approximated by tractable oes that preserve the relevat iformatio about the system performace. As i most multi-class queueig systems, the queue legth of the licesed users, who have a higher priority, will vaish asymptotically. This is ot a problem if the queue legth of the licesed users does ot affect the users i service i a asymptotic sese, which is the case i most applicatios, ad oe ca still obtai the maagerial isights by aalyzig the limit of scaled processes aloe. However, whether the umber of waitig licese users is asymptotically small or exactly zero is importat i our settig as it determies whether a ulicesed user should vacate a chael but the scaled processes fail to preserve such importat iformatio. Thus, the aalysis requires iformatio from both scaled ad uscaled processes, ivolves trackig the two processes simultaeously, ad eeds to use the averagig priciple. These requiremets are rare i the literature with oly a few exceptios such as Perry ad Whitt (20a), Luo ad Zhag (203), Pag ad Perry (205). I this sectio, we first itroduce our fluid model x(t) = (z (t), q (t), z 2 (t), q 2 (t)) which will be used to approximate the stochastic process X (t) = (Z (t), Q (t), Z 2 (t), Q 2 (t)) i our system with the justificatios to be provided i Sectio 5. We the derive the steady-state performace ad study the optimal sharig decisios of the system i the steady state usig the fluid approximatios.

12 2 4.. The Fluid Model Defiitio 2 (FLUID MODEL). The process x(t) = (z (t), q (t), z 2 (t), q 2 (t)) evolves accordig to the costrait ad the followig differetial equatios 0 = [ z (t) z 2 (t)]q (t), (3) z (t) = [ β(t)]λ + α(t)[ z (t) + µz 2 (t)] z (t), (4) q (t) = β(t)λ α(t)[ z (t) + µz 2 (t)], (5) z 2(t) = [ β(t)][λ 2 + θq 2 (t)] [p + α(t)( p)]µz 2 (t), (6) q 2(t) = ( φ)β(t)[λ 2 + θq 2 (t)] + ( φ)α(t)( p)µz 2 (t) θq 2 (t), (7) where β(t) ad α(t) deped o how costrait (3) is met. If q (t) > 0, the β(t) = α(t) = ; if z (t) + z 2 (t) <, the β(t) = α(t) = 0; otherwise, { ( ) + [λ + λ 2 + θq 2 (t) z (t) pµz 2 (t)][ z (t) + µz 2 (t)] β(t) = mi, }, (8) [λ + λ 2 + θq 2 (t)][ z (t) + µz 2 (t)] λ [ z (t) + pµz 2 (t)] { } λ β(t) α(t) = mi z (t) + µz 2 (t),. (9) The fluid model defied above is built o the evolutio of the system described i Sectio 3. As we will explai i Sectio 5 ad defie formally i Appedix B, β(t) is the istataeous delay probability of the licesed users ad α(t) is the istataeous probability that a ulicesed user has to release the chael after a service sessio (i.e., there are waitig licesed users i the system), referred to as the iterruptio probability, uder the fluid model. Thus, the differetial equatios (4) (7) are quite ituitive. Take equatio (4) for a example. The rate of icrease i z (t) cosists of two parts: () Whe the licesed users arrive (at the rate λ ), there is a available chael (with probability β(t)); (2) Whe the licesed users fiish service (at the rate z (t)) or the ulicesed users fiish a service sessio (at the rate µz 2 (t)); there are waitig licesed users (with probability α(t)). The rate of decrease i z (t) is z (t), which is the rate the licesed users occupyig the chaels fiish service. For equatio (7), the rate of icrease i q 2 (t) cosists of two parts: () Whe the ulicesed users arrive or those i the orbit queue perform sesig (at the rate λ 2 + θq 2 (t)), they fid all chaels occupied (with probability β(t)) but decide ot to abado the system (with probability φ); (2) Whe the ulicesed users fiish a service sessio (at the rate µz 2 (t)), they eed aother oe (with probability p) ad fid licesed users waitig (with probability α(t)) but do ot abado the system (with probability φ). The rate of decrease i q 2 (t) is θq 2 (t), which is the rate the ulicesed users i queue sese for available chaels.

13 Whe z (t)+z 2 (t) < or q (t) > 0, the system dyamics is quite simple ad resembles that of the may-server queues i call ceter applicatios. For example, whe z (t) + z 2 (t) <, the differetial equatios (4) (7) reduce to q (t) 0 ad z (t) = λ z (t), z 2(t) = λ 2 + θq 2 (t) pµz 2 (t), q 2(t) = θq 2 (t). Otherwise, the system dyamics is more complicated. Moreover, the process x( ) ca move back ad forth amog differet cases, which makes the aalysis eve more challegig as show i Appedix A. Despite the complexity, the fluid model ca be solved umerically. Furthermore, we ca obtai the steady state of the fluid model i Theorem to approximate the steady state of the origial system. For example, β := lim β(t) ad T H 2 := lim pµz 2 (t) ca be used to accurately approximate t t the steady-state delay probability of the licesed users ad the throughput rate of the ulicesed users, respectively. Note that fluid models fail to yield probabilistic performace measures i most applicatios. Similar to Gurvich ad Perry (202), our fluid model actually provides accurate approximatios for them The Steady State of the Fluid Model While the offered load of such a system is λ + λ 2 µ 2, the effective load is edogeous as the average time for which a ulicesed user occupies a chael µ defied i (2) depeds o the decisio µ t. Sice p is the average umber of service sessios eeded to fulfill the service requiremet of a ulicesed user, the effective service time of a ulicesed user is. Thus, the effective load of pµ the system is λ + λ 2 pµ, where pµ = µ 2µ s µ t +µ s. Note that the effective load is always o less tha the offered load ad equals the offered load if ad oly if there is o restrictio o the service process of the ulicesed users, i.e., µ t =. The shorter the trasmissio time i a service sessio, the more service sessios (ad hece sesig) are eeded for the ulicesed users to complete their trasmissios ad the more cogested the system is. Depedig o the effective load of the system, the steady state of the fluid limits are give i the ext theorem whose proof ca be foud i Appedix A. Theorem. There exists a uique solutio to the fluid model. Moreover, the limitig behavior of the fluid model as t ca be characterized as follows. A vector-valued fuctio x(t) is called a solutio of the fluid model if it is absolutely cotiuous o every closed time iterval ad satisfies equatios (4) (7) almost everywhere. 3

14 4 (. If λ + λ 2 >, the lim x(t) = λ pµ t, 0, λ (β, α) is the uique solutio to, φ θφ λ [ λ 2 pµ ( λ ) ]), T H 2 = pµ ( λ ), ad α = λ + µ ( )β, (0) λ ( p)α γ = β + ( β) p + ( p)α, () λ 2 E[γ K ] = λ 2 pµ ( λ ), (2) where K follows a geometric distributio with parameter φ. ( 2. If λ + λ 2, the lim x(t) = λ pµ t, 0, λ 2 ),, 0 T H pµ 2 = λ 2, ad α = β = 0. We first describe the ituitio behid the delay probability β i Equatios (0) (2) before discussig the steady-state behavior i more detail i the ext sectio. Equatio (0) is obtaied by pluggig lim x(t) ito (9). Note that β is also the probability that a ulicesed user will t be served upo arrival or after each sesig activity while waitig i the orbit queue, ad ( p)α p+( p)α is the probability that a ulicesed user i service will be iterrupted. Thus, γ i () is the probability that a ulicesed user will experiece blockage or iterruptio ad hece eeds to decide whether or ot to abado the system at least oce. Sice K represets the umber of times a ulicesed user eeds to decide whether to abado the system, E[γ K ] is the probability that a ulicesed user will abado the system. So the left-had side of (2) ca be uderstood as the abadomet rate of the ulicesed users, while the right-had side is also the abadomet ( ) rate but calculated by subtractig the rate p λ at which ulicesed users complete their service from the total arrival rate λ 2. Give that E[γ K ] = γφ, we actually have a closed-form γ( φ) expressio (see (29) i Appedix A) for the delay probability β from solvig (0) (2). Table also presets a compariso betwee the simulated delay probability ad throughput rate ad the approximatio based o the fluid model. As oe ca see, the fluid approximatio works well, especially whe is large, which is the case i our applicatio. Furthermore, our simulatio also reveals that the average queue legth of the licesed users is ideed quite short (vaishes asymptotically). For istace, the 0.95 cofidece iterval of the queue legth of the licesed users is 0.072±.000 with determiistic times ad 0.02±.000 with expoetial times whe = 4000 ad µ t = 0.6. From Theorem, we ca see that the system performace is isesitive to θ, the frequecy at which the ulicesed users sese for a available chael while waitig i the orbit queue. This is because, although θ affects the trasiet of the differetial equatios (4) (7), it iflueces the steady state through the total sesig speed θq 2 (i.e., whe the derivatives of the left had side equal 0). As θ icreases, the ulicesed users are allowed to sese chael availability more

15 frequetly ad hece abado the system sooer, which lowers the umber of waitig ulicesed users q 2. It turs out that, uder such a mechaism, the total sesig speed remais costat as θ varies. The isesitivity of θ o the system performace is further cofirmed by a simulatio study i Appedix C. Thus, the decisio o the sesig frequecy should be based o techological (e.g., power cosumptio as sesig cosumes power) rather tha operatioal cocers. Whe the system is effectively uder or critically loaded, i which case the offered load is λ + λ 2 µ 2, all users will be served without delay i the steady state ad o restrictio eeds to be imposed o the ulicesed users. Whe the system is effectively overloaded, i which case the ( ) offered load may or may ot be above, oly p λ of the ulicesed users will fiish service per uit time ad the ulicesed users will experiece iterferece with a positive probability. Theorem also reveals some iterestig steady-state behavior that differs from that of the fluid models i most applicatios such as call ceters.. It is well uderstood i the queueig literature that, if a system is critically loaded, there is a positive probability that delay will occur, eve with a extra capacity of O( λ ) i most o-preemptive models i applicatios such as call ceters (see Halfi ad Whitt 98). I our applicatio, the requiremet for ulicesed users to sese chael availability while waitig i the orbit queue guaratees the availability of idle chaels for all licesed users upo arrival eve whe the system is critically loaded, leadig to a zero delay probability for licesed users asymptotically. We ote a similar result i Pag ad Perry (205) that, by cotrollig whe outboud calls ca be made, reservig a logarithmic order umber of servers i a call ceter ca achieve a zero delay probability for iboud calls asymptotically whe the system is critically loaded. 2. It is also well uderstood that, whe a system is overloaded, customers will experiece delay almost surely i most call ceter applicatios because all servers are busy all the time (see Whitt 2006). I our applicatio, however, a arrivig licesed user still has a chace to eter service upo arrival eve whe there is a large umber of ulicesed users i the orbit queue as it takes time for them to sese chael availability. Hece, the delay probability of the licesed users, which is edogeously determied by the load through (0) (2), is strictly less tha. Eve if a licesed user is delayed upo arrival, his waitig time is i the order of ( ) O, which is relatively short but may still be sigificat i data trasmissio. λ Essetially, the restrictio that the ulicesed users are ot allowed to sese chael availability costatly makes the system operate more like a preemptive oe for the licesed users tha a o-preemptive oe. 5

16 Sesitivity of the System Performace By Theorem, sesig frequecy does ot affect the system performace. Thus, we will focus o the impact of the legth of trasmissio time µ t (or equivaletly the legth of the service sessio) o the throughput of the ulicesed users T H 2 ad the delay probability of the licesed users β. Corollary. T H 2 is always icreasig i the trasmissio time µ t, i.e., allowig the ulicesed users loger service sessios will icrease the system throughput rate. β is quasi-covex i µ t. More specifically, If λ + λ 2 µ 2 or µ s µ 2 ( λ λ ) 2 + µ ( 2 λ λ ) µ 2, the β decreases i µ t (see Figure 2(a) (b)). Otherwise, there exists a threshold ˆµ t < such that β decreases i µ t whe µ t ˆµ t ad icreases i µ t whe µ t > ˆµ t (see Figure 2(c) (d)). Figure 2 illustrates the delay probability as a fuctio of the trasmissio time for various λ, λ 2 ad µ s whe = µ 2 =, θ = 0.4 ad φ = 0.5. Note that the purpose of restrictig the amout of time the ulicesed users ca occupy a chael is to limit the iterferece of the ulicesed users to the service of the licesed users. Thus, ituitively, shorter service sessios should always lead to a lower delay probability. The corollary reveals that this is true oly if ˆµ t = 0 which happes whe the workload from both types of users are high eough ad the sesig time is moderate (Figure 2(d)). Whe the system is overloaded ad sesig is ot too time cosumig, imposig too short service sessios will oly icrease the effective load ad hece the delay probability, while imposig relatively loger service sessios will icrease the delay probability as expected (Figure 2(c)). Whe the system is uder or critically loaded, shorter service sessios will either have o impact o the delay probability or tur the system ito a effectively overloaded oe, icreasig the delay probability (Figure 2(a)). Whe the system is overloaded ad sesig takes a log time, it oly makes sese to allow a ulicesed user to trasmit for a sigificat amout of time i order to lower the delay probability (Figure 2(b)) Optimal Sharig Decisios i the Steady State I this sectio, we ivestigate whether a give bad of spectrum should be shared with ulicesed users ad the trasmissio time µ t that maximizes the throughput rate of the ulicesed users while keepig the delay probability of the licesed users below a certai level, η. Note that θ does ot affect the system performace by Theorem ad the trasmissio time is the oly decisio. Furthermore, maximizig the throughput rate of the ulicesed users is equivalet to maximizig the throughput rate of the system sice the throughput rate of the licesed users is a costat.

17 7 β β λ 2 = λ 2 = µ t µ t (a) Uder (λ = 0.2, λ 2 = 0.79) ad critically (λ = 0.2, λ 2 = 0.8) loaded, µ s = 0.00 (b) Overloaded (λ = 0.2, λ 2 = 0.9) ad µ s = 0.0 > β µ t β µ t (c) Overloaded (λ = 0.2, λ 2 = 0.9) ad µ s = < Figure 2 The delay probability as a fuctio of the trasmissio time (d) Overloaded (λ = 0.8, λ 2 =.7) ad µ s = 0.25 [0.23, 0.4] Whether ad How to Share Whe the system is uder or critically loaded, the system may also be effectively overloaded if oe allocates shorter service sessios to the ulicesed users. However, by Theorem, eve if the ulicesed users are allowed to trasmit for as log as they eed, the delay probability coverges to zero ad all users are able to complete their trasmissio without delay as. Thus, we do ot eed to restrict the service sessio of the ulicesed users whe is large eough. Whe the system is overloaded, it is also effectively overloaded regardless of the legth of the ( ) allocated service sessio. By Theorem, T H 2 = p λ ad β is the solutio to (2). Thus, the optimizatio problem ca be writte as max µ t 0 pµ (3)

18 8 s.t. β η, µ = (µ 2 + µ t )µ s µ t + µ s, p = µ 2 µ 2 + µ t. Sice the objective fuctio is icreasig i µ t, the optimizatio problem reduces to oe of fidig the largest µ t that satisfies the delay costrait. Whe η is so small that the feasible regio is empty, o ulicesed users should be allowed i the system. Oce η is large eough to make the feasible regio o-empty, ulicesed users will be allowed i the system. As η icreases, the optimal trasmissio time icreases. The optimal trasmissio time =, i.e., the ulicesed µ t µ t users are allowed to complete their trasmissio oce they start occupyig a chael, if η is larger tha the poit such that the feasible regio becomes ubouded. Figure 3 demostrates the optimal spectrum sharig decisio as a fuctio of η ad λ 2 whe λ = 0.2, = µ 2 =, µ s = 0.00, θ = 0.4 ad φ = 0.5. The upper curve specifies the arrival rate above which the ulicesed users should ot be allowed to share the spectrum ad the lower oe is the threshold below which there is o eed to restrict the service sessio of the ulicesed users, i.e., =. µ t Sice our aalysis oly holds i a asymptotic sese (as the umber of chaels becomes large), there is still a o-egligible delay probability whe the system is uder or critically loaded ad is ot sufficietly large. For the same example i Figure 3 with λ = 0.2, Figure 4 demostrates the optimal sharig decisios, obtaied through simulatio, as a fuctio of η ad λ 2 for = 00, 200, 500, 000, i which case the system is uder or critically loaded whe λ As oe ca see, the structure of the optimal sharig decisio remais the same, ad as icreases, sharig is more likely to occur ad the ulicesed users should be allowed loger service sessios Sesitivity of the Optimal Decisio The optimal decisio ad the throughput µ t rate of the ulicesed users T H2 deped o the system parameters i the followig way. decreases, i.e., the ulicesed users are allowed a shorter tras- Propositio. The optimal µ t missio time, as () λ icreases while keepig λ + λ 2 costat whe = µ 2 ; (2) λ ad decrease while keepig λ costat; ad (3) λ 2 ad µ 2 icrease while keepig λ 2 µ 2 costat. Furthermore, the optimal throughput T H2 will decrease uder () ad (2). Note that uder all the scearios, the total offered load λ + λ 2 µ 2 is kept costat. Propositio states that shorter service sessios should be allocated to the ulicesed users () as the proportio

19 λ η Figure 3 The optimal sharig decisio as a fuctio of η ad λ 2 for a overloaded system of licesed users icreases whe all users have idetical service requiremets, (2) if there are fewer licesed users but with loger service times; ad (3) if there are more ulicesed users but with shorter service times. While () ad (3) are more ituitive, (2) holds because the delay probability oly depeds o both λ ad λ. A delay icidet of a licesed user is couted as oe regardless of his service requiremet. With fewer licesed users, each delay cotributes more to the delay probability ad it is easy to show that shorter service sessios should be imposed o the ulicesed users. ) ( ) As a result, the optimal throughput rate pµ ( λ = µ 2µ s λ µ t +µs decreases uder scearios () ad (2) as expected. These suggest that spectrum sharig is beeficial to systems with a smaller proportio of licesed users or a large umber of licesed users with shorter service times. Uder sceario (3), although shorter service sessios have a egative impact o the throughput rate due to more sesig activities, the icrease i the umber of ulicesed users with shorter service times has a positive impact. Thus, the impact o throughput rate is ot mootoe.

20 do ot share λ do ot share share with restrictio λ share with restrictio share without restrictio 0.7 share without restrictio η η (a) = 00 (b) = do ot share share with restrictio 0.8 do ot share share with restrictio λ 2 λ share without restrictio 0.75 share without restrictio 0.7 Figure η (c) = 500 The optimal sharig decisios as a fuctio of η ad λ η (d) = Justificatios for the Fluid Approximatio I this sectio, we demostrate i Theorem 2 that the scaled process X (t) coverges to the fluid model x(t) i Sectio 4. Sice the proof of the theorem is quite ivolved, we describe the mai ideas of the proof through the costructio of the fluid model, especially the istataeous delay probability of the licesed users β(t) ad the istataeous iterruptio probability of the ulicesed users α(t). The complete proof ca be foud i Appedix B. For ay T > 0, let D([0, T ], R 4 ) be the space of all right-cotiuous R 4 valued fuctios o [0, T ] with left limits, edowed with the Skorohod J topology. Let deote covergece i distributio for radom objects i R 4 equipped with Euclidia topology or D([0, T ], R 4 ) with Skorohod J topology. Theorem 2 (Fluid Approximatio). Uder the asymptotic regime, if X (0) x(0) as, the X (t) x(t) i D([0, T ], R 4 ), where x(t) is the fluid model specified i Defiitio 2. Need for both scaled ad uscaled processes If we let Λ i (t) deote the Poisso process with the rate λ i, the Λ (t + δ) Λ (t) is the total umber of licesed users arrivig i a small

21 2 iterval [t, t + δ], amog which t+δ t {I (s )=0}dΛ (s) will fid o idle chaels upo arrival ad have to wait. Thus, the delay probability of the licesed users durig this small time iterval [ t+δ t {I (s )=0} dλ (s) is E ]. That is, the delay probability depeds o the iformatio about the Λ (t+δ) Λ (t) uscaled process I (t) 0 as it determies whether a ulicesed user i service should vacate a chael at the ed of a service sessio. Likewise, we eed to keep track of the uscaled process of the queue legth of the licesed users Q (t) ad obtai the probability of a ulicesed user i service beig iterrupted i [t, t + δ]. However, I (t) 0 vaishes i the asymptotic regime alog with the process Q (t) 0 as i most systems with multiple classes ad we eed to keep track of both the scaled ad uscaled processes i order to obtai the system dyamics ad asymptotic system performace. The system dyamics usig the averagig priciple To obtai the system dyamics, we eed to apply the averagig priciple by first expressig the probabilities i [t, t + δ] as a time average usig PASTA (Poisso arrivals see time average). For istace, the fractio of time for which there is o idle chael i the system is Let δ t+δ t {I (s )=0}ds = δ t+δ {I (t+ s )=0}ds. (4) t m (t) = Q (t) I (t). (5) We study the system dyamics for the uscaled process m ( t + s ) for 0 s δ. Note that the process m ( t + ) oscillates aroud zero i the order of. Whe m ( t + s ) < 0 (there are idle chaels ad o licesed users waitig by ()), the process icreases by whe there is a ew arrival at the rate λ + λ 2 or oe of the ulicesed users i the orbit queue eters service after sesig the system at the rate θ Q 2 (t + s ). The process decreases by whe a user (licesed or ulicesed) completes service at the rate Z (t + s ) + pµ Z 2 (t + s ). Whe ( m t + ) s > 0 (there are licesed users waitig ad o idle chaels), the process icreases by at the rate λ ad decreases at the rate Z (t + s ) + µ Z 2 (t + s ). We refer readers to Appedix B. for the detailed system dyamics. It is the log-ru average behavior of m ( t + ) that plays the key role i determiig the fractio i (4) whe becomes large i the asymptotic regime. Explaatio for β(t) ad α(t) As oe ca see, the process m ( t + ) is ot a Markov process sice its evolutio depeds o a higher dimesio process tha itself. However, if we approximate the above metioed rates by their fluid couterparts, i.e., Z i (t + s ) by z i(t), Q 2 (t + s ) by q 2(t), ad λ i by λ i, we have a Markov process as i Figure 5 whose steady-state distributio π t ca be easily obtaied. We use π t (j) to approximate the asymptotic proportio of time for which there are j licesed users i the queue whe j > 0 ad there are j idle chaels whe j < 0. The delay